# Topological Completeness is Weakly Hereditary

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## Theorem

Let $T = \struct {S, \tau}$ be a topological space which is topologically complete.

Let $V \subseteq S$ be a closed subspace of $T$.

Then $V$ is also topologically complete.

That is, topological completeness is weakly hereditary.

## Proof

## Sources

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces